Stopped Sigma-Algebra preserves Inequality between Stopping Times

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Theorem

Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a filtered probability space.

Let $T$ and $S$ be stopping times with respect to $\sequence {\FF_n}_{n \ge 0}$ such that:

$\map S \omega \le \map T \omega$

for each $\omega \in \Omega$.

Let $\FF_S$ and $\FF_T$ be the stopped $\sigma$-algebras associated with $S$ and $T$ respectively.


Then:

$\FF_S \subseteq \FF_T$


Proof

Let $A \in \FF_S$ and $t \in \Z_{\ge 0}$.

If $\omega \in \Omega$ is such that:

$\map T \omega \le t$

we have:

$\map S \omega \le t$

So:

$\set {\omega \in \Omega : \map T \omega \le t} \subseteq \set {\omega \in \Omega : \map S \omega \le t}$

for each $t \in \Z_{\ge 0}$.

So, from Intersection with Subset is Subset, we have:

$\set {\omega \in \Omega : \map T \omega \le t} \cap \set {\omega \in \Omega : \map S \omega \le t} = \set {\omega \in \Omega : \map T \omega \le t}$

So, we have from Intersection is Associative:

$A \cap \set {\omega \in \Omega : \map T \omega \le t} = \paren {A \cap \set {\omega \in \Omega : \map S \omega \le t} } \cap \set {\omega \in \Omega : \map T \omega \le t}$

Since $T$ is a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$ we have:

$\set {\omega \in \Omega : \map T \omega \le t} \in \FF_t$

and since $A \in \FF_S$ we have:

$A \cap \set {\omega \in \Omega : \map S \omega \le t} \in \FF_t$

Since $\FF_t$ is closed under finite intersection, we have:

$A \cap \set {\omega \in \Omega : \map T \omega \le t} \in \FF_t$

for all $t \in \Z_{\ge 0}$.

So $A \in \FF_T$, giving $\FF_S \subseteq \FF_T$.

$\blacksquare$


Sources

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